Price and variance of anarchy in mean-variance cost density-shaping stochastic differential games

This paper introduces the Variance of Anarchy (VoA) metric to compliment existing measures of efficiency loss in dynamic games due to decentralized mechanisms. The VoA is inspired by Price of Anarchy (PoA) and Price of Information (PoI) measures that have been used previously in the literature. We propose a new design procedure for decentralized control algorithms using PoA and VoA that identifies the optimal control solution for competing agents among a family of decentralized controllers by solving an optimization of a summed PoA and VoA objective function over a parameter space. The design method is illustrated with a stochastic model for queue server dynamics and two separate optimal control problems - the first involving noncooperative agents, and the second a team. For each problem, a family of density-shaping cumulant controls is computed corresponding to a parametric target cumulant set, and the optimal chosen via this new procedure. Simulation results are provided to compare the controller to a baseline 2CC control.